Final Exam
MAT103 : Calculus I (Fall 2025)
(150 minutes)
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Problem 1. (15 points) Given a function f (x) and a value x = a, evaluate lim f (x) or explain
x→a
why it does not exist. Justify your answers using the limit laws. In each case explain what your
limit calculation tells you about the graph of f .
2
at a = 2.
(i) f (x) = 2xx2 −x−2
−5x+2
3 x
e )
(ii) f (x) = sin(2x
at a = 0.
2x tan3 (x)
4
)
as a → ∞.
(iii) f (x) = sin(x
x2 +x4
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Problem 2. (15 points)
(i) There are points of the graph of f (x) = x4 − 2x2 + 4x where the tangent line is both parallel
to y = 4x and lies underneath the graph. Find the x-coordinates of those points.
(ii) Find an expression for the slope function of f (x) = ln(x)x .
(iii) A curve C is defined by the equation 2ex−y−1 + x = 0. Find an equation of the tangent line
to C at the point (x0 , y0 ) = (−2, −3) and use it to estimate the value of a when the point
(a, −3.3) is on the curve C.
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Problem 3. (15 points)
(i) Evaluate the following definite integrals:
Z 4√
Z π
x+1
sin(|x|)
(a)
dx
(b)
dx
x
3
1
−π
Z 1
(c)
1/2
√
2dx
1 − x2
(ii) A “triangular” region in the xy-plane is bounded above by y = ln(x), below by x + y = 1,
and on the right by the line x = e.
(a) Sketch the region.
(b) Set up a definite integral with respect to x that computes the area of this region.
(c) Find the area of the region using integration with respect to y and geometric reasoning.
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Problem 4. (15 points) In this problem we study the function
f (x) = |ex − e2x |.
(i) What is the domain of f and what asymptotes does its graph have?
(ii) Where is f continuous and where is it differentiable?
(iii) Find and classify all local and global extrema of f . For any global extrema, explain why
they are global.
(iv) Sketch the graph of f .
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Problem 5. (15 points) A lab ships a liquid reagent in bottles. The manufacturing cost per
bottle, each of which holds v liters, is
Cm (v) = 2 + 3v
dollars. The shipping company charges by dimensional weight and the lab’s packaging standard
forces the packaging box to have side-length proportional to v 1/3 . As a result the shipping cost
per bottle (in dollars) is
6v 2/3 ,
0<v≤1
Cs (v) =
.
3/2
2v + v + 3, v > 1
(i) The lab sells the reagent at a fixed price of $18/liter. Find a formula for the profit per
bottle P (v).
(ii) Is the function P (v) that you found in part (i) differentiable at v = 1? Explain why.
(iii) Find the value of v that maximizes profit. Explain why it is a global maximum.
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Blank page (can be used as extra work space).
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